On the Asymptotic Behavior of Advanced Differential Equations with a Non-Canonical Operator

Omar Bazighifan 1,2,† and Ioannis Dassios 3,*,† 1 Department of Mathematics, Faculty of Science, Hadhramout University, Hadhramout 50512, Yemen; o.bazighifan@gmail.com 2 Department of Mathematics, Faculty of Education, Seiyun University, Hadhramout 50512, Yemen 3 AMPSAS, University College Dublin, Dublin 4, Ireland * Correspondence: ioannis.dassios@ucd.ie † These authors contributed equally to this work.


Introduction
In recent decades, many authors have studied problems of a number of different classes of advanced differential equations including the asymptotic and oscillatory behavior of their solutions, see [1][2][3][4][5][6][7][8] and the references cited therein. For some more recent oscillation results, see [9][10][11][12][13][14][15][16][17][18][19][20]. The interest in studying advanced differential equations is also caused by the fact that they appear in models of several areas in science. In [21][22][23], singular systems of differential equations are used to study the dynamics and stability properties of electrical power systems. Some additional mathematical background on this can be found in [24]. Systems of differential equations with delays are used to study additional properties of electrical power systems in [25,26]. Non-linear advanced differential equations can be used to describe complex dynamical networks, see [27][28][29], and bring new insight to their stability. Furthermore, this type of equations can be also used in the modeling of dynamical networks of interacting free-bodies, see [30]. Finally, properties of advanced differential equations are used in the study of singular differential equations of fractional order, see [31,32]. Several other examples in Physics can be found in [33]. In this paper, we consider an even-order non-linear advanced differential equation with a non-canonical operator of the following type: where υ ≥ υ 0 , κ is even and β is a quotient of odd positive integers. The operator L y is said to be in canonical form if ∞ υ 0 a −1/β (s) ds = ∞; otherwise, it is called noncanonical. Throughout this work, we suppose that: C3: g ∈ C (R, R) such that g (x) /x β ≥ k > 0, for x = 0 and under the condition , a (υ) > 0 and y (υ) satisfies (1) on [υ y , ∞).

Definition 2. Let
A kernel function H i ∈ C (D, R) is said to belong to the function class , written by H ∈ , if, for i = 1, 2, has a continuous and nonpositive partial derivative ∂H i /∂s on D 0 and there exist functions and Next we will discuss the results in [34][35][36]. Actually, our purpose in this article is to complement and improve these results. Agarwal et al. in [34,35] studied the even-order nonlinear advanced differential equations By means of the Riccati transformation technique, the authors established some oscillation criteria of (5). Grace and Lalli [36] investigated the second-order neutral Emden-Fowler delay dynamic equations y (κ) (υ) + q (υ) y (η (υ)) = 0, and established some new oscillation for (5) under the condition To prove this, we apply the previous results to the equation if we set κ = 4 and λ = 2, then by applying conditions in [34][35][36] on Equation (8), we find the results in [35] improves those in [36]. Moreover, the those in [34] improves results in [35,36]. Thus, the motivation in our paper is to complement and improve results in [34][35][36]. We will use the following methods: Method of comparison with second-order differential equations.

Lemma 3 ([2]
). Let β be a ratio of two odd numbers, V > 0 and U are constants. Then

Lemma 4.
Suppose that y is an eventually positive solution of (1). Then, there exist three possible cases:
Let case (S 3 ) hold. By recalling that a (υ) y (κ−1) (υ) β is non-increasing, we obtain Dividing the latter inequality by a 1/β (s) and integrating the resulting inequality from υ to u, we get Letting u → ∞, we obtain Furthermore, we get due to (19). Now define we see that φ (υ) < 0 for υ ≥ υ 1 , and It follows from (1) and (19) that From Lemma 2, we find Thus, we have From (22), we obtain From [37], we can see that Equation (11) is non-oscillatory, which is a contradiction. Theorem 1 is proved.
Based on the above results and Theorem 1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with β = 1.
In the next theorem, we employ the integral averaging technique to establish a Philos-type oscillation criteria for (1): and, lim sup Then every solution of (1) is either oscillatory or satisfies lim υ→∞ y (υ) = 0.
For this we leave the results to researchers interested .

Conclusions
In this article we studied we provided three new Theorems on the oscillatory and asymptotic behavior of a class of even-order advanced differential equations with a non-canonical operator in the form of (1).

Author Contributions:
The authors claim to have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.